1 / 35

On the Complexity of Trial and Error

On the Complexity of Trial and Error. Shengyu Zhang Joint work with Xiaohui Bei and Ning Chen. STOC’13, also at arXiv:1205.1183. Motivating example: drug development. Known vs. unknown.

baby
Download Presentation

On the Complexity of Trial and Error

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. On the Complexity of Trial and Error Shengyu Zhang Joint work with XiaohuiBeiand NingChen STOC’13, also at arXiv:1205.1183

  2. Motivating example: drug development

  3. Known vs. unknown • If the virus (its DNA sequence, chemical composition, …) and its interactions to human body are known, then the reagent is much easier to find. • Unfortunately, often • the virus causing a pandemic is largely unidentified. • human body is too complicated a system to fully understand • Learn it first? Efficiency!

  4. Standard method for drug development: Trial and error • allergic reaction, • severe headache, • …

  5. Phenomena • Task: search for a solution (reagents) to satisfy a bunch of constraints (no side effect). • Input (virus, human body): unknown. • Allowed operations: solution testing. • Approach: trial-and-error. • Efficiency is crucial • preferably better than learning the unknown input. • Next: More examples.

  6. Normal-Form Game Prisoner’s dilemma Players have to choose an action anyway • Issue: In many scenarios, players do not know their payoffs. • Eg.: When a new business model appears, companies don’t really know what strategies will give how much payoff.

  7. Stable Matching 7

  8. Stable Matching Issue: Individuals may not know their preferences exactly. An assignment has to be determined “… a systematic and continuous approach to fitting the right person to the right job at the right time has long been the Holy Grail of workforce organization.” --- McKinsey Quarterly, 2003

  9. In a broad sense

  10. In a broad sense

  11. Questions to study

  12. Input space Solution space Model CSP A: Verification oracle V Algorithm • If more than one violations, then V returns an arbitrary one. • Worst-case analysis. • We usually don’t know how Nature returns a violation. • We’re only given the index, but not the content of . • We usually get an error signal, but don’t know the exact reason of the error. (e.g. “headache”: don’t know which ingredients of the reagent cause the problem.)

  13. Input space Solution space Model CSP A: Verification oracle V Algorithm Question: What if Ais already hard? (Then Au couldn’t be easy. What to ask?) 13

  14. Input space Solution space Model CSP A: Verification oracle V • Computation oracle A How much extradifficulty is introduced due to the lack of input knowledge? Algorithm • Add another computation oracle that computes Aitself. • Au, A: unknown-input version of Ais not much harder. 14

  15. formula assignments Example: SAT Verification oracle V Computation oracle SAT Algorithm

  16. More detailed comparisons: 3 pages in paper. Related Models SAT: Deciding the satisfiability and finding a solution is easy, Learning input formula is hard. (Even finding a formula with the same solution set needs exponential time.)

  17. Results I: Positive • Message 1: Despite the very little information provided by V, there are efficient algorithms for many natural problems.

  18. Results I: Positive • The following problems are no harder than known-input case. • Namely, • Nash: Find a Nash equilibrium of a normal-form game • Core: Find a core of a cooperative game • Stable matching: Find a stable matching a two-sided matching market (with agents) • . • SAT: Find a satisfying assignment of a CNF formula (with m clauses and n variables) • if and if .

  19. Results II: negative • Message 2: Not all problems with unknown inputs are easy---lack of input information does impose more difficulty for some problems.

  20. Relation to traditional complexity theory: Hardness of GroupIsou= hardness of NP-complete. Results II: Negative • The following are much harder than their known-input case. • Graph Isomorphism: Find isomorphism between two graphs. • , i.e.  trial-efficient algorithm. • The algorithm works if given computation oracleSAT • GraphIsou • Group Isomorphism: Find isomorphism between two groups. • , and algorithm works with SAT orcle. • GroupIsou • Subset sum: Find a partition (of given numbers) with equal sum • , i.e. even trial-complexity is exponentially high.

  21. Nash Equilibrium CSP: (Note: Infinite number of constraints) • Theorem: There is a polynomial-time algorithm, when equipped with the computation oracle solving the know-input Nash problem.

  22. Basic Idea: Ellipsoid Method • Non-linear (as function of variables and ) • The solution space is not convex • And… we haven’t used the computation oracle yet

  23. One more (critical) issue: • Degenerated search space: a single point • Cannot use standard perturbation approach • Use a “strong separation orcle” machinery by Grötschel, Lovász, and Schrijver, 1988. Nash Equilibrium Idea: Search for the input –a single point (convex!) thanks to the fact that NE always exist! • If it returns A separation oracle! • But since is an NE for

  24. Stable Matching • CSP: or , • Theorem. There is a polynomial-time randomized algorithm with trials • Theorem. Any randomized algorithm needs queries, regardless of time. • Approach: upper bound---Find the input. Lower bound---probabilistic method. • Idea (for algorithm): Reduce to sorting.

  25. Sorting • CSP: iff, • Lemma.  a poly-time randomized algorithm A solving Sortu with trials. • Note: This  a poly-time algorithm B solving StableMatchingu with trials. StableMatchingu V >1 >1’ … … >n >n’ Gale-Shapley π

  26. Sorting • CSP: iff, • Lemma.  a poly-time randomized algorithm A solving Sortu with O(n log n) trials. • (Theorem. This is tight: Any randomized algorithm needs Ω(n log n) trials.) • Both algorithm and lower bound uses order theory. • Average height : average rank of (i.e., # of s.t.) in all linear extensions • Theorem[Kahn, Saks, 1984] : . • Algorighm: propose order according to • Any violation, say , would cut (by Kahn-Saks) a constant fraction of possible linear extensions.

  27. Sorting • One more issue: is #P-hard to compute. [Brightwell, Winkler, 1991] • Fortunately, there is a fully polynomial randomized approximation scheme (FPRAS) to count the number of linear extensions [Dyer, Frieze, Kannan, 1989] • Can be used to approximate to within 0.5 in polynomial time. • Then we can use Kahn-Saks. (This time we really need < 1.) • Lemma.  a poly-time randomized algorithm A solving Sortu with O(n log n) trials. • (Theorem. This is tight: Any randomized algorithm needs Ω(n log n) trials.) • Both algorithm and lower bound uses order theory. • Average height : average rank of (i.e., # of s.t.) in all linear extensions • Theorem[Kahn, Saks, 1984] : . • Algorighm: propose order according to • Any violation, say , would cut (by Kahn-Saks) a constant fraction of possible linear extensions! Open question: min # of trials of deterministic and poly-time algorithms for Sortingu and StableMatchingu?

  28. Group Isomorphism CSP: ,

  29. Group Isomorphism • All we need to do is to find a avoiding existing forbidden pairs of triples. • Note: Groups are defined by these triple structures! • So, with the help of GroupIsooracle, it’s possible to exploit group structures to achieve this. What if Group Isomorphism itself is given as the computation oracle?

  30. Group Isomorphism • All we need to do is to find a avoiding existing forbidden pairs of triples. • Note: Groups are defined by these triple structures! • So, with the help of GroupIsooracle, it’s possible to exploit group structures to achieve this. What if Group Isomorphism itself is given as the computation oracle? Theorem. GroupIsou is NP-complete. Approach: Reduction to Hamiltonian Cycle finding.

  31. Reduction • HamCycleFinding: Given a -node graph with a HamCycle, find a HamCycle HamCycle, define cyclic group Finding an iso of ⇒ find a HamCycle in • Algorithm A (solving GroupIsou.) Issue: We don’t know the cycle (and thus T)! Neither does A! (Algorithm on unknown inputs!)

  32. Reduction • HamCycleFinding: Given graph H w/ p nodes and a HamCycle, find a HamCycle HamCycle, define cyclic group Finding an iso of ⇒ find a HamCycle in • Algorithm A (solving GroupIsou.) S V Simulator S: poly-time Real V: unaffordable We probably lose something… Correctness!

  33. Reduction • HamCycleFinding: Given graph H w/ p nodes and a HamCycle, find a HamCycle HamCycle, define cyclic group Finding an iso of ⇒ find a HamCycle in • Algorithm A (solving GroupIsou.) S V Simulator S: poly-time Real V: unaffordable Property: The first time S gives a wrong answer to A, we’ve found an Hamiltonian cycle in !

  34. Summary • Set up a framework for studying CSP problems • Reshape the complexity of problems: • Easy: Nash, Core, StableMatching, SAT • Hard: GroupIso, GraphIso To unknown inputs: To trial-and-error: • From art to science To learning theory: • Solving instead of learning. • Hopefully a supplement. • Tells when we need to look at input structure. • SAT: No need. • GraphIso: Necessary. To complexity theory?

  35. Thanks

More Related